A Level Further Maths CP2 Methods in Differential Equations

nature of roots of auxiliary equation, z and m, do what?

determine the general solution to the 2nd order differential equation a(d²y/dx²) + b(dy/dx) + c = 0

to find the general solution to the differential equation a(d²y/dx²) + b(dy/dx) + cy = f(x)

- solve corresponding homogeneous equation a(d²y/dx²) + b(dy/dx) + cy = 0 to find the complementary function (C.F.) -choose appropriate form for the particular integral, P.I., and substitute into the original equation to find the values of any coefficients -general solution is y = C.F. + P.I.

first case of nature of auxiliary equation roots

1: b² > 4ac auxiliary equation has two real roots α and β, (α≠β) general solution will be of the form y = Ae^(αx) + Be^(βx) where A and B are arbitrary constants

second case of auxiliary equation roots

2: b² = 4ac auxiliary equation has one repeated root α general solution will be of the form y = (A+Bx)e^(αx) where A and B are arbitrary constants

third case of auxiliary equation roots

3: b² < 4ac auxiliary equation has 2 complex conjugate roots α and β equal to p±qi general solution will be of the form y = e^(px)(Acosqx + Bsinqx) where A and B are arbitrary constants

particular integral is what

a function which satisfies the original differential equation

can solve a first-order differential equation of the form dy/dx + P(x)y = Q(x) by

multiply by the integrating factor e^(∫P(x)dx)